What is a research mathematician? How is it even possible to do research in mathematics? We’ll examine these questions through the mathematics and philosophy of Jacques Hadamard (1865–1963). Hadamard’s model of mathematical invention (preparation, incubation, illumination, verification) will guide our exploration of mathematical discovery. We’ll learn that a research mathematician is a combination of problem-solver, nitpicker, big-picture thinker, experimental scientist, aesthetician and dreamer.